From generating functions to the geometric Binder cumulant

We present an overview of the role of generating functions in quantum mechanical contexts, mainly in the modern theory of polarization and in the study of quantum phase transitions. Generating functions enable the derivation of moments and cumulants, quantities which characterize the fluctuations of an underlying probability distribution. In all of the cases we review, the fluctuations are those of a quantum system. We show that the original formalism for geometric phases, in which a quantum system is taken around an adiabatic cycle, can be extended to the case when degeneracy points are encountered along the cycle (quasiadiabatic cycles). The essential tool for this extension is a generalized Bargmann invariant which plays the role of a generating function. From the cumulants generated this way one can form ratios according to the Binder cumulant scheme in statistical mechanics. Such geometric Binder cumulants are sensitive to gap closure, as such, they are useful in identifying metal-insulator, localization, and quantum phase transitions. We present example calculations on simple model systems whose localization properties are well known to validate to approach. We also complement our geometric Binder cumulant calculations with results for the fidelity susceptibility, a quantity directly related to the quantum geometry of the parameter space. CONTENTS I. Introduction 2 II. Basic ingredients 4 A. The problem of the polarization in crystalline systems 4 B. Generating functions, moments, cumulants 5 C. The excess kurtosis and the Binder cumulant 6 D. Berry phases 7 1. The Berry phase 7 2. The open path Berry phase (Zak phase) 8 3. The single point Berry phase 9 E. The generating function in quantum geometry 9 III. Generating functions in crystalline systems 9 A. Periodic probability distributions 10 B. The modern polarization theory for crystalline many-body systems 11 C. Gauge invariant cumulants 12 D. Generating functions for the Berry phase 12 E. Constructing the geometric Binder cumulant for quantum cycles 14 IV. Demonstrative examples 14 A. The Fermi sea 15 B. The Su-Schrieffer-Heeger model 16 C. Localization in the Aubry-André model 17 D. The Aubry-André transition through the fidelity suscebtibility 19 V. Conclusion 19.